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I am looking for solution of the following: Is it possible to rewrite constraint

$\frac{a^T x}{c^T x + d} \leq K$,

where $x$ is vector variable and $K$ is scalar variable as well

as a convex constraint, i.e so it doesn't contain any multiplications. So far my considerations are that function on the left is quasiconvex(rational linear function), therefore it is possible to make a level set on it. However, the current form of the constraint is not implementable. Most likely that I have to think out about substitution, but can't work it out so far. Thanks.

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  • $\begingroup$ Do you know anything about whether $K$ and $c^{T}x+d$ are positive? $\endgroup$ – Brian Borchers Oct 19 '16 at 5:08
  • $\begingroup$ Yes, both $K$ and $c^T x +d$ are positive $\endgroup$ – Mykola Servetnyk Oct 19 '16 at 5:09
  • $\begingroup$ Then why don't you multiply both sides by $c^T x+d$? $\endgroup$ – polfosol Oct 19 '16 at 5:17
  • $\begingroup$ Because K is a variable, so that on the right side we will have $K(c^T x + d)$, so it is multiplication of the variables. $\endgroup$ – Mykola Servetnyk Oct 19 '16 at 5:19
  • $\begingroup$ You either have multiplication of variables or division of variables. Generally you can't scape both $\endgroup$ – polfosol Oct 19 '16 at 5:37

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